Grade 8 Chapter 7: Rational and Irrational Numbers In this chapter we first review the real line model for numbers, as discussed in Chapter 2 of seventh grade, by recalling how the integers and then the rational numbers are associated to points in the line. Having associated a point on the real line to every rational number, we ask the question, do all points correspond to a rational number? Recall that a point on the line is identified with the length of the line segment from the origin to that point (which is negative if the point is to the left of the origin). Through constructions (given by \tilted" squares), we make an observation first made by the Pythagorean society 2500 years ago that there are lengths (such as the diagonal of a square with side length 1) that do not correspond to a rational number. The constructionp produces numbers whose squares are integers; leading us to introduce thep symbol A to represent a number whose square is A. We also introduce the cube root 3 V to represent the side length of a cube whose volume is V . The technique of tilted squares provides an opportunity to observe the Pythagorean theorem: a2 + b2 = c2, where a and b are the lengths of the legs of a right triangle, and c is the length of the hypotenuse. In the nextp section we return to the construction of a square of area 2, and show that its side length ( 2) cannot be equal to a fraction, so its length is not a rationalp number. We call such a number an irrational number. The same argument works for 5 and other lengths constructed by tilted squares.p It is a fact that if N is an integer, either itp is a perfect square (the square of an integer), or N is not a quotient of integers; that is N is an irrational number. In the next section we turn to the question: can we represent lengths that are not quotients of integers, somehow by numbers? The ancient Greeks were not able to do this, due mostly to the lack of an appropriate system of expressing lengths by their numerical measure. For us today, this effective system is that of the decimal representation of numbers (reviewed in Chapter 1 of seventh grade). We recall that by dividing an interval into ten equal parts, we move on to the next decimal point. To be precise: Given a point x on the real line count the number of unit interval that can be put completely in the interval from the origin to x. This is the integral part of the number associated to x (negative if x is left of the origin). Now divide the unit interval into ten equal parts, and count how many fit in the space left over. That digit goes in the tenth place. repeating this for as many times as necessary, we can associate to every (terminating) decimal a point on the line. A rational number is represented by a terminating decimal only if the denominator is a product of twos and fives. Thus many rational numbers (like 1/3,1/7, 1/12,...) are not represented by terminating decimals, but they are represented by repeating decimals, and similarly, repeating decimals represent rational numbers. We now view the decimal expansion of a number as providing an algorithm for getting as close as we please to its representing point on the line through repeated subdivisions by tenths. In fact, every decimal expansion represents a point on the line, and thus a number, and unless the decimal expansion is 1 terminating or repeating, it is irrational. The question now becomes: can we represent all lengths by decimal expansions? We start with square roots, and illustrate Newton's method for approximating square roots: Start with some reasonable estimate, and follow with the recursion 1 N anew = aold + : 2 aold Through examples, we see that this method produces the decimal expansion of the square root of N to any required degree of accuracy. Finally, we point out that to do arithmetic operations with irrational as well as rational numbers, we have to be careful: to get within a specified number of decimal points of accuracy we may need much better accuracy for the original numbers. Section 7.1. Representing numbers geometrically Interpret numbers geometrically through the real line. First, let us recall how to represent the rational number system by points on a line. With a straight edge, draw a horizontal line. Given any two points a and b on the line, we say that a < b if a is to the left of b. The piece of the line between a and b is called the interval between a and b. It is important to notice that for two different points a and b we must have either a < b or b < a. Also, recall that if a < b we may also write this as b > a. Pick a point on a horizontal line, mark it and call it the origin, denoted by 0. Now place a ruler with its left end at 0. Pick another point (this may be the 1 cm or 1 in point on the ruler) to the right of 0 and denote it as 1.We also say that the length of the interval between 0 and 1 is one )pr one unit. Mark the same distance to the right of 1, and designate that endpoint as 2. Continuing on in this way we san associate to each positive number a point on the line. Now mark off a succession of equally spaced points on the line that lie to the left of 0 and denote them consecutively as −1; − 2; − 3;::: . In this way we can imagine all integers placed on the line. We can associate a half integer to the midpoint of any interval; so that the midpoint of the interval between 3 and 4 is 3.5, and the midpoint of the interval between -7 and -6 is -6.5. If we divide the unit interval into three parts, then the first part is a length corresponding to 1/3, the first and second parts correspond to 2/3, and indeed, for any integer p, by putting p copies end to and on the real line (on the right of the origin is p > 0, and on the left if p < 0), we get to the length representing p=3. We can replace 3 by any positive integer q, by constructing a length which is one qth of the unit interval. In this way we can identify every rational number p=q with a point on the horizontal line, to the left of the origin if p=q is negative, and to the right if positive. Now draw a vertical line through the origin, and do the same on that line, using the same interval as the unit interval. Now, to every pair of rational numbers (a; b) we can associate a 2 point in the plane: Go along the horizontal (the x-axis) to the point a. Now, set the pins of a protractor at the endpoints of the interval from the origin to the point b, and measure this out on the vertical line through a, arriving at the point to be identified with (a; b). Finally, we can measure the length of any line segment on the plane using a protractor, as follows: set the pins of a protractor at the endpoints of the line segment and then measure that length along the horizontal (or vertical) axis, with one pin at the origin. The other pin lands on or near some point representing a fraction p=q. The closest such fraction is an estimate for the length of the line segment. Example 1. In figure 1 the unit lengths are half an inch each. What are the lengths of the sides of the triangle? Activity a) On a coordinate plane like that of figure 1, draw any 8 7 C 6 5 4 3 B 2 1 A O 1 2 3 4 5 6 7 8 Figure 1 triangle and measure the lengths of its sides. What can you say about the lengths of the sides of the triangle? Now draw a triangle with one side horizontal, and the other vertical. Is there anything more that you can say about the lengths of the sides. Now, it is important to know that, by using a ruler we can always estimate the length of a line segment by a fraction ( a rational number), and the accuracy of the estimate depends upon the detail of our ruler. The question we now want to raise is this: can any length be realized by a rational number ? Construct square roots using tilted squares. Observe the pythagorean theorem in all of these 3 tiled squares. Explain a proof of the Pythagorean Theorem and its converse. (informally) 8G6 The coordinate system on the plane provides us with the ability to assign lengths to line segments. Let us review this: Figure 2 Example 2. In Figure 2 we have drawn a tilted square (dashed sides) within a horizontal square. If each of the small squares bounded in a solid line is a unit square (the side length is one unit), then the area of the entire figure is 2 × 2 = 4. The dashed, tilted square is composed of precisely half (in area) of each of the unit squares, since each of the triangles outside the tilted square corresponds to a triangle inside the tilted square.